I would like to see a clear, rigorous and elementary proof of the following statement:

Let X be a (not necessary quasi-projective, separated) algebraic variety over the complex numbers, and let U,V be two affine open subsets of X. Then the intersection of U and V is affine.

Does the proof change if one substitutes "scheme" for "variety"?

separatedness. For non-separated schemes, it'll be wildly unture (there is a weaker notion of quasiseparatedness which means that intersections of open affines arefinite unionsof open affines, but there are schemes that don't even have this property.) – Ilya Grigoriev Mar 18 '10 at 19:44